Variance and Standard Deviation Cheat Sheet

Variance and Standard Deviation Cheat Sheet

Quick Revision

Variance and standard deviation both measure how far a set of values spreads out from its mean. The standard deviation is simply the square root of the variance, so it is in the same units as the data.

1. The Mean

Mean (x̄) = Σx / n

Every measure of spread is taken about this mean.

2. Variance

Variance (σ²) = Σ(x − x̄)² / n

Shortcut form, which avoids the deviations:

σ² = (Σx² / n) − x̄²

3. Standard Deviation

Standard deviation (σ) = √(variance)

A small σ means the values cluster near the mean; a large σ means they are widely scattered.

4. Effect of Changing the Data

• Add a constant k to every value → the mean rises by k, but σ is unchanged.
• Multiply every value by k → the mean and σ are both multiplied by |k|, and the variance by k².

5. Worked Example

For the data 2, 4, 6, 8:

Mean      = (2 + 4 + 6 + 8) / 4 = 5
Deviations = −3, −1, 1, 3  →  squares 9, 1, 1, 9  →  sum 20
Variance  = 20 / 4 = 5
SD        = √5 ≈ 2.24

6. Exam Quick Tips

• Variance is never negative — it is an average of squares.
• The standard deviation carries the same unit as the data; the variance does not.
• Adding the same number to every value does not change the spread.

Keep revising with the rest of the Maths section:

• All Maths topics
• Statistics Cheat Sheet
• Average Cheat Sheet

How this was made: Drafted with AI assistance; spot-checked by the editor on 31 October 2026.